Fourth-order compact solution of the nonlinear Klein-Gordon equation

Numerical Algorithms

Volumn 52, Issue 4, 2009, Pages 523-540

  Dehghan, Mehdi ^a   Mohebbi, Akbar ^a,b   Asgari, Zohreh ^c  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b UNIVERSITY OF KASHAN   (Iran)

^c ALZAHRA UNIVERSITY   (Iran)

Author keywords

Compact finite difference; Diagonally implicit Runge Kutta Nystr m (DIRKN) methods; High accuracy; Klein Gordon (KG) equation; Soliton

Indexed keywords

EID: 76149122486     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-009-9296-x     Document Type: Article

References (34)

1. Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71, 16-30 (2006).
2. Bratsos, A. G., Twizell, E. H.: The solution of the sine-Gordon equation using the method of lines. Int. J. Comput. Math. 61, 271-292 (1996).
3. Bratsos, A. G.: On the numerical solution of the Klein-Gordon equation. Numer. Methods Partial Differ. Equ. 25, 939-951 (2009).
4. Bratsos, A. G.: A third order numerical scheme for the two-dimensional sine-Gordon equation. Math. Comput. Simul. 76, 271-282 (2007).
5. Bratsos, A. G.: The solution of the two-dimensional sine-Gordon equation using the method of lines. J. Comput. Appl. Math. 206, 251-277 (2007).
6. Bratsos, A. G.: A numerical method for the one-dimensional sine-Gordon equation. Numer. Methods Partial Differ. Equ. 24, 833-844 (2008).
7. Butcher, J. C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008).
8. Caudrey, P. J., Eilbeck, I. C., Gibbon, J. D.: The sine-Gordon equation as a model classical field theory. Nuovo Cim. 25, 497-511 (1975).
9. Deeba, E. Y., Khuri, S. A.: A decomposition method for solving the nonlinear Klein-Gordon equation. J. Comput. Phys. 124, 442-448 (1996).
10. Dehghan, M., Shokri, A.: Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. J. Comput. Appl. Math. (2009, in press).
11. Denga, X., Zhaob, M., Lic, X.: Travelling wave solutions for a nonlinear variant of the PHI-four equation. Math. Comput. Model. 49, 617-622 (2009).
12. Dodd, R. K., Eilbeck, I. C., Gibbon, J. D., Morris, H. C.: Solitons and Nonlinear Wave Equations. Academic, London (1982).
13. El-Sayed, S. M.: The decomposition method for studying the Klein-Gordon equation. Chaos Solitons Fractals 18, 1025-1030 (2003).
14. Franco, J. M., Gómez, I.: Accuracy and linear stability of RKN methods for solving second-order stiff problems. Appl. Numer. Math. 59, 959-975 (2009).
15. Guo, B. Y., Li, X., Vázquez, L.: A Legendre spectral method for solving the nonlinear Klein- Gordon equation. Mat. Appl. Comput. 15, 19-36 (1996).
16. Inc, M., Ergüt, M., Evans, D. J.: An efficient approach to the Klein-Gordon equation: an application of the decomposition method. Int. J. Simul. Process Model. 2, 20-24 (2006).
17. Jiménez, S., Vázquez, L.: Analysis of four numerical schemes for a nonlinear Klein-Gordon equation. Appl. Math. Comput. 35, 61-94 (1990).
18. Lee, I. J.: Numerical solution for non-linear Klein-Gordon equation by collocation method with respect to spectral method. J. Korean Math. Soc. 32, 541-551 (1995).
19. Li, S., Vu-Quoc, L.: Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klien-Gorodn equation. SIAM J. Numer. Anal. 32, 1839-1875 (1995).
20. Liao, W., Zhu, J., Khaliq, A. Q. M.: A fourth-oder compact algorithm for nonlinear reaction-diffusion equations with Neumann boundary conditions. Numer. Methods Partial Differ. Equ. 22, 600-616 (2005).
21. Lindblad, H., Rodnianski, I.: The weak null condition for Einstein's equations. C. R. Math. Acad. Sci. Paris 336, 901-906 (2003).
22. Lynch, M. A. M.: Large amplitude instability in finite difference approximations to the Klein-Gordon equation. Appl. Numer. Math. 31, 173-182 (1999).
23. Mohebbi, A., Dehghan, M.: The use of compact boundary value method for the solution of two-dimensional Schrödinger equation. J. Comput. Appl. Math. 225, 124-134 (2009).
24. Sassaman, R., Biswas, A.: Soliton perturbation theory for phi-four model and nonlinear Klein-Gordon equations. Commun. Nonlinear Sci. Numer. Simulat. 14, 3239-3249 (2009).
25. Scott, A. C.: A nonlinear Klein-Gordon equation. Am. J. Phys. 37, 52-61 (1969).
26. Shakeri, F., Dehghan, M.: Numerical solution of the Klein-Gordon equation via He's variational iteration method. Nonlinear Dyn. 51, 89-97 (2008).
27. Spotz, W. F.: High-order compact finite difference schemes for computational mechanics. Ph. D. Thesis, University of Texas at Austin, Austin, TX (1995).
28. Strauss, W. A., Vázquez, L.: Numerical solution of a nonlinear Klein-Gordon equation. J. Comput. Phys. 28, 271-278 (1978).
29. Wazwaz, A. M.: Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations. Chaos Solitons Fractals 28, 1005-1013 (2006).
30. Wazwaz, A. M.: New travelling wave solutions to the Boussinesq and the Klein-Gordon equations. Commun. Nonlinear Sci. Numer. Simulat. 13, 889-901 (2008).
31. Wong, Y. S., Chang, Q., Gong, L.: An initial-boundary value problem of a Klein-Gordon equation. Appl. Math. Comput. 84, 77-93 (1997).
32. Dehghan, M., Mirezaei, D.: Numerical solution to the unsteady two-dimensional Schrodinger equation using meshless local boundary integral equation method. Int. J. Numer. Methods Eng. 76, 501-520 (2008).
33. Dehghan, M., Shokri, A.: A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math. Comput. Simul. 79, 700-715 (2008).
34. Dehghan, M., Mirzaei, D.: The dual reciprocity boundary element method (DRBEM) for twodimensional sine-Gordon equation. Comput. Method Appl. Mech. Eng. 197, 476-486 (2008).